Skiba points in free end-time problems

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Abstract

Since the end of the seventies Skiba points have been studied in infinite time optimal control problems with multiple steady states. At such a Skiba point the decision maker is indifferent between choosing trajectories that approach different steady states. This paper extends this theory towards free end-time optimal control problems, where the decision maker collects a salvage value at the endogenous horizon date. In particular, besides operating forever, the decision maker can choose to stop operations immediately, or to operate during a finite time interval after which the decision maker stops and collects a salvage value.

This paper illustrates the new theory with a firm-level capital accumulation problem in which the manager has the option to sell the firm (be acquired) at any time. This situation may be relevant for certain high-tech start-ups that create intellectual property which may be of value to an acquiring firm and which is hard for an outside firm to tap in other ways (e.g., via licensing).

Introduction

Sethi (1977), Sethi (1979), Skiba (1978), Dechert and Nishimura (1983) established the existence of indifference points, or Skiba points,1 in infinite time optimal control models with multiple steady states. At a Skiba point the decision maker is indifferent between choosing trajectories that approach different steady states.

The existence of Skiba points implies that the optimal solution is history dependent, i.e. which long run steady state will be reached depends on the initial value of the state variable. For policy makers such knowledge is especially important when one steady state can be designated as a “bad” equilibrium, while the other one is a “good” equilibrium. Then, by implementing the right policy measure, a policy maker can shift the Skiba point in such a way that the “good” equilibrium will be reached. Papers on this topic include (Matsuyama, 1991, Hartl and Kort, 1996) on an environmental economics application, and Caulkins et al. (2013) on rational addiction.

This paper extends the notion of Skiba points to the class of optimal control problems in which the decision maker can stop operations at any time and collect a salvage value. In such a framework essentially three qualitatively different solution candidates can be distinguished. Besides operating forever, the decision maker can decide to stop operations immediately and collect the salvage value, or to operate for a finite time and then stop. We show that Skiba points exist where the decision maker is indifferent (1) between “operating forever” and “operating for a finite time and then stopping”, and (2) between “operating forever” and “stopping immediately”.

We focus on Skiba points where the decision maker is indifferent between the policies “operate forever” or “operate for a finite time and collect the salvage value at the endogenous horizon date” since then both options involve dynamics. One might expect that in a one-dimensional model, if one policy involves departing the Skiba point by increasing the state variable, then for the other policy the state must start to decrease. We prove that is true – almost always. There is a hairline case in which both policies can move away from the Skiba point in the same direction, but in that case they apply the same control and all points traversed are Skiba points; the two alternate solutions move from one Skiba point to another Skiba point until reaching a point where one is indifferent between continuing to operate forever and ceasing operation immediately.

Recent contributions in the area of multiple optimal solutions include (Kato et al., 2007, Kiseleva and Wagener, 2010, Ernst and Semmler, 2010, Zeiler et al., 2010, Akao et al., 2011).

We illustrate our results by studying the problem of a company whose management has the option to sell the firm (be acquired). This problem could be relevant with regard to high growth, technology-oriented companies. Some such companies remain independent, and some grow to become very profitable (e.g. Google whose founders are now billionaires, Apple, Facebook, and Adobe Systems). Yet, for every Google, there are many other startups that make their founders rich in a different way, by allowing the founders to grow the technology for a time, but then be acquired voluntarily. Examples are Hotmail, Android, YouTube and PayPal (acquired by Microsoft, Google and eBay).

While there are many capital accumulation models (e.g., Jorgenson, 1963, Gould, 1968, Barucci, 1998, Feichtinger et al., 2006), to our knowledge none focus on the possibility that it could be optimal that a startup may want to be acquired by another (usually larger) firm.

The remainder of the paper is organized as follows. Section 2 shows for a general framework that whenever Skiba points exist where the decision maker is indifferent between a policy of operating forever or ceasing operation at a finite point of time, then if for the policy of staying alive forever the state variable starts by growing, exactly the opposite happens when the decision maker chooses to stop operations in finite time, and vice versa. Section 3 illustrates this theory by analyzing the capital accumulation model of the firm where the manager has the option to sell it at any time. Section 4 concludes.

Section snippets

Skiba points in a free end-time problem

This section characterizes the optimal paths in one-state free end-time problems, specifically when a Skiba point exists where the decision maker is indifferent between “operating forever” and “operating during a finite time interval” after which the decision maker collects the salvage value. At such a Skiba point it holds that if the state variable starts growing under one policy, it starts by shrinking for the other policy (see also Fig. 3, Fig. 5, Fig. 7 corresponding to our illustrative

The model of the firm

We start from a concave revenue function version of the capital accumulation model by Barucci (1998) and Hartl and Kort (2000), Hartl and Kort (2004). We extend this model by introducing the option that the start-up firm can be taken over by another, established company.

The firm starts with some goodwill which we model as a fixed constant, m, as it is not the focus of the dynamics under investigation here. In addition the firm has a capital stock, K, that it exploits to produce revenue and in

Conclusion

The starting point of this paper is the literature on Skiba points, including the early contributions by Sethi (1977), Sethi (1979), Skiba (1978), Dechert and Nishimura (1983). This literature considers infinite time optimal control models with multiple steady states. A Skiba point is defined such that right at this point the decision maker is indifferent between choosing trajectories that approach different steady states. This paper generalizes the Skiba point literature to models with a free

Acknowledgements

The authors like to thank the editor, three anonymous reviewers and Vladimir Veliov for their helpful comments. This research was supported by the Austrian Science Fund (FWF) under Grant P21410-G16 and P23084-N13.

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